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1. Revisiting closed asymptotic couples

   https://doi.org/10.1017/S0013091522000219  

   Aschenbrenner, Matthias; van den Dries, Lou; van der Hoeven, Joris (May 2022, Proceedings of the Edinburgh Mathematical Society)

   Abstract Every discrete definable subset of a closed asymptotic couple with ordered scalar field $${\boldsymbol {k}}$$ is shown to be contained in a finite-dimensional $${\boldsymbol {k}}$$ -linear subspace of that couple. It follows that the differential-valued field $$\mathbb {T}$$ of transseries induces more structure on its value group than what is definable in its asymptotic couple equipped with its scalar multiplication by real numbers, where this asymptotic couple is construed as a two-sorted structure with $$\mathbb {R}$$ as the underlying set for the second sort. 

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2. Informative and non-informative decomposition of turbulent flow fields

   https://doi.org/10.1017/jfm.2024.1007  

   Arranz, Gonzalo; Lozano-Durán, Adrián (December 2024, Journal of Fluid Mechanics)

   Not all the information in a turbulent field is relevant for understanding particular regions or variables in the flow. Here, we present a method for decomposing a source field into its informative$$\boldsymbol {\varPhi }_{I}(\boldsymbol {x},t)$$and residual$$\boldsymbol {\varPhi }_{R}(\boldsymbol {x},t)$$components relative to another target field. The method is referred to as informative and non-informative decomposition (IND). All the necessary information for physical understanding, reduced-order modelling and control of the target variable is contained in$$\boldsymbol {\varPhi }_{I}(\boldsymbol {x},t)$$, whereas$$\boldsymbol {\varPhi }_{R}(\boldsymbol {x},t)$$offers no substantial utility in these contexts. The decomposition is formulated as an optimisation problem that seeks to maximise the time-lagged mutual information of the informative component with the target variable while minimising the mutual information with the residual component. The method is applied to extract the informative and residual components of the velocity field in a turbulent channel flow, using the wall shear stress as the target variable. We demonstrate the utility of IND in three scenarios: (i) physical insight into the effect of the velocity fluctuations on the wall shear stress; (ii) prediction of the wall shear stress using velocities far from the wall; and (iii) development of control strategies for drag reduction in a turbulent channel flow using opposition control. In case (i), IND reveals that the informative velocity related to wall shear stress consists of wall-attached high- and low-velocity streaks, collocated with regions of vertical motions and weak spanwise velocity. This informative structure is embedded within a larger-scale streak–roll structure of residual velocity, which bears no information about the wall shear stress. In case (ii), the best-performing model for predicting wall shear stress is a convolutional neural network that uses the informative component of the velocity as input, while the residual velocity component provides no predictive capabilities. Finally, in case (iii), we demonstrate that the informative component of the wall-normal velocity is closely linked to the observability of the target variable and holds the essential information needed to develop successful control strategies. 

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3. Phoretic motion in active matter

   https://doi.org/10.1017/jfm.2021.530  

   Brady, John F. (September 2021, Journal of Fluid Mechanics)

   A new continuum perspective for phoretic motion is developed that is applicable to particles of any shape in ‘microstructured’ fluids such as a suspension of solute or bath particles. Using the reciprocal theorem for Stokes flow it is shown that the local osmotic pressure of the solute adjacent to the phoretic particle generates a thrust force (via a ‘slip’ velocity) which is balanced by the hydrodynamic drag such that there is no net force on the body. For a suspension of passive Brownian bath particles this perspective recovers the classical result for the phoretic velocity owing to an imposed concentration gradient. In a bath of active particles that self-propel with characteristic speed $$U_0$$ for a time $$\tau _R$$ and then change direction randomly, taking a step of size $$\ell = U_0 \tau _R$$ , at high activity the phoretic velocity is $$\boldsymbol {U} \sim - U_0 \ell \boldsymbol {\nabla } \phi _b$$ , where $$\phi _b$$ is a measure of the ‘volume’ fraction of the active bath particles. The phoretic velocity is independent of the size of the phoretic particle and of the viscosity of the suspending fluid. Because active systems are inherently out of equilibrium, phoretic motion can occur even without an imposed concentration gradient. It is shown that at high activity when the run length varies spatially, net phoretic motion results in $$\boldsymbol {U} \sim - \phi _b U_0 \boldsymbol {\nabla } \ell$$ . These two behaviours are special cases of the more general result that phoretic motion arises from a gradient in the swim pressure of active matter. Finally, it is shown that a field that orients (but does not propel) the active particles results in a phoretic velocity $$\boldsymbol {U} \sim - \phi _b U_0 \ell \boldsymbol {\nabla }\varPsi$$ , where $$\varPsi$$ is the (non-dimensional) potential associated with the field. 

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4. Momentum-Based Variance Reduction in Non-Convex SGD

   Cutkosky, Ashok; Orabona, Francesco (January 2019, Advances in neural information processing systems)

   Wallach, H.; Larochelle, H.; Beygelzimer, A.; d'Alché-Buc, F.; Fox, E.; Garnett, R. (Ed.)

   Variance reduction has emerged in recent years as a strong competitor to stochastic gradient descent in non-convex problems, providing the first algorithms to improve upon the converge rate of stochastic gradient descent for finding first-order critical points. However, variance reduction techniques typically require carefully tuned learning rates and willingness to use excessively large "mega-batches" in order to achieve their improved results. We present a new algorithm, STORM, that does not require any batches and makes use of adaptive learning rates, enabling simpler implementation and less hyperparameter tuning. Our technique for removing the batches uses a variant of momentum to achieve variance reduction in non-convex optimization. On smooth losses $$F$$, STORM finds a point $$\boldsymbol{x}$$ with $$E[\|\nabla F(\boldsymbol{x})\|]\le O(1/\sqrt{T}+\sigma^{1/3}/T^{1/3})$$ in $$T$$ iterations with $$\sigma^2$$ variance in the gradients, matching the optimal rate and without requiring knowledge of $$\sigma$$. 

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5. A simple analytic model for predicting the wicking velocity in micropillar arrays

   https://doi.org/10.1038/s41598-019-56361-7  

   Krishnan, Siva Rama; Bal, John; Putnam, Shawn A. (December 2019, Scientific Reports)

   Abstract Hemiwicking is the phenomena where a liquid wets a textured surface beyond its intrinsic wetting length due to capillary action and imbibition. In this work, we derive a simple analytical model for hemiwicking in micropillar arrays. The model is based on the combined effects of capillary action dictated by interfacial and intermolecular pressures gradients within the curved liquid meniscus and fluid drag from the pillars at ultra-low Reynolds numbers$${\boldsymbol{(}}{{\bf{10}}}^{{\boldsymbol{-}}{\bf{7}}}{\boldsymbol{\lesssim }}{\bf{Re}}{\boldsymbol{\lesssim }}{{\bf{10}}}^{{\boldsymbol{-}}{\bf{3}}}{\boldsymbol{)}}$$ $\left({10}^{-7}\lesssim \mathrm{Re}\lesssim {10}^{-3}\right)$. Fluid drag is conceptualized via a critical Reynolds number:$${\bf{Re}}{\boldsymbol{=}}\frac{{{\bf{v}}}_{{\bf{0}}}{{\bf{x}}}_{{\bf{0}}}}{{\boldsymbol{\nu }}}$$ $\mathrm{Re}=\frac{v_0{x}_0}{\nu }$, wherev₀corresponds to the maximum wetting speed on a flat, dry surface andx₀is the extension length of the liquid meniscus that drives the bulk fluid toward the adsorbed thin-film region. The model is validated with wicking experiments on different hemiwicking surfaces in conjunction withv₀andx₀measurements using Water$${\boldsymbol{(}}{{\bf{v}}}_{{\bf{0}}}{\boldsymbol{\approx }}{\bf{2}}\,{\bf{m}}{\boldsymbol{/}}{\bf{s}}{\boldsymbol{,}}\,{\bf{25}}\,{\boldsymbol{\mu }}{\bf{m}}{\boldsymbol{\lesssim }}{{\bf{x}}}_{{\bf{0}}}{\boldsymbol{\lesssim }}{\bf{28}}\,{\boldsymbol{\mu }}{\bf{m}}{\boldsymbol{)}}$$ $\left(v_0\approx 2m/s,25\mu m\lesssim x_0\lesssim 28\mu m\right)$, viscous FC-70$${\boldsymbol{(}}{{\boldsymbol{v}}}_{{\bf{0}}}{\boldsymbol{\approx }}{\bf{0.3}}\,{\bf{m}}{\boldsymbol{/}}{\bf{s}}{\boldsymbol{,}}\,{\bf{18.6}}\,{\boldsymbol{\mu }}{\bf{m}}{\boldsymbol{\lesssim }}{{\boldsymbol{x}}}_{{\bf{0}}}{\boldsymbol{\lesssim }}{\bf{38.6}}\,{\boldsymbol{\mu }}{\bf{m}}{\boldsymbol{)}}$$ $\left(v_0\approx 0.3m/s,18.6\mu m\lesssim x_0\lesssim 38.6\mu m\right)$and lower viscosity Ethanol$${\boldsymbol{(}}{{\boldsymbol{v}}}_{{\bf{0}}}{\boldsymbol{\approx }}{\bf{1.2}}\,{\bf{m}}{\boldsymbol{/}}{\bf{s}}{\boldsymbol{,}}\,{\bf{11.8}}\,{\boldsymbol{\mu }}{\bf{m}}{\boldsymbol{\lesssim }}{{\bf{x}}}_{{\bf{0}}}{\boldsymbol{\lesssim }}{\bf{33.3}}\,{\boldsymbol{\mu }}{\bf{m}}{\boldsymbol{)}}$$ $\left(v_0\approx 1.2m/s,11.8\mu m\lesssim x_0\lesssim 33.3\mu m\right)$. 

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